包郵復形和Cohen-Macaulay性質(英文版)

作者：武同鎖//郭錦

出版社：科學出版社出版時間：2021-01-01

開本： 16開 頁數： 254

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復形和Cohen-Macaulay性質(英文版) 版權信息

ISBN：9787030703026
條形碼：9787030703026 ; 978-7-03-070302-6
裝幀：一般膠版紙
冊數：暫無
重量：暫無
所屬分類：
自然科學
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數學

復形和Cohen-Macaulay性質(英文版) 內容簡介

本書包含組合交換代數近幾年來的一些主要研究成果，選題圍繞單純復形、代數復形以及Cohen-Macaulay性質展開，其中的CM性質是交換代數中很為核心的研究課題。全書共分為7章。

復形和Cohen-Macaulay性質(英文版) 目錄

Contents
Preface
Notations
Chapter 1　Preliminaries on Cohen-Macaulay Rings and Modules　1
1.1　Jacobson radical and NAK Lemma　2
1.2　Modules with finite lengths　3
1.3　On graded rings and minimal graded free solution　6
1.4　Cohen-Macaulay rings and Cohen-Macaulay Modules　11
1.4.1　Dimension， height and Krull’s PI theorem　11
1.4.2　Depth Lemma and depthI(M)　14
1.4.3　Local rings: Krull dimension and a system of parameters　17
1.4.4　Cohen-Macaulay modules and Cohen-Macaulay rings: local case　22
1.4.5　Cohen-Macaulay rings: non-local case　26
1.4.6　Cohen-Macaulay rings: graded case　28
1.4. 7　Gorenstein rings　34
1.5　Sequentially Cohen-Macaulay modu1es　35
Chapter 2　Abstract Simplicial Complexes　40
2.1　Definitions， fundamiental properties and examples　40
2.1.1　Abstract simplex and abstract simplicial complex＊40
2.1.2　0ther notations and symbols on a simplicial complex ＊　41
2.1.3　Fundaｍental operations on sub-complexes and geometric realization of an abstract simplicial complex　42
2.2　The facet idea1I(*) and Stanley-Reisner ideal I*of a simplicial complex *　47
2.2.1　Monomial ideals and ideal operations　47
2.2.2　The Stanley-Reisner (nonface） ideal I* and facet ideal I*　47
2.2.3　The Alexander dual simplicial complex * of*and related properties　48
2.2.4　Square-free monomial ideal I: its nonface complex， facet complex; I* and f-ideals　54
2.3　Relative simplicial complexes and relative nonface ideals　55
Chapter 3　Shellable Simplicial Complexes　57
3.1　Destriction and examples　57
3.2　Restriction maps and Rearrangement Lemmas　60
3.3　(r，s)-skeleton *　64
3.4　Shifted， vertex-decomposable and shellable conditions for a simplicial complex　66
3.5　Shellable and k-decomposable　69
Chapter 4　Chain Complex Reduced from a Simplicial Complex and Koszul Complexes　73
4.1　The chain complex reduced from an abstract simplicial complex and reduced homology groups　74
4.2　Koszul complexes of lengths 1 or 2　79
4.3　Koszul complexes of geueral length　80
4.3.1　Exterior algebra constructed from a module　80
4.3.2　Koszul complexes： two commonly used definitions　81
4.4　Koszul complexes: a S田nmary of main results　83
4.5　Other resolutions and complexes of monomial ideals　84
4.5.1　The Taylor resolution　84
4.5.2　The Scarf complex　86
4.5.3　The Lyubeznik resolutions　89
Chapter 5　(Sequentially) Cohen-Macaulay Simplicial Complexes and Graphs　91
5.1　Cohen-Macaulay simplicial complexes　92
5.1.1　Fundamental properties and characterizations　92
5.1.2　Connected in codimension one　97
5.1.3　Minimal Cohen-Macaulay simplicial complexes and shelled over　100
5.2　Matroid complexes　104
5.3　Pure shellable， constructible， and Cohen-Macaulay　107
5.4　A graded ideal with linear quotients and shellable complexes　111
5.4.1　A graded id兇i with linear quotients　111
5.4.2　Shellable complexes and monomial ideals having linear quotients　115
5.4.3　Powers of edge ideals of graphs and regularity　118
5.4.4　A polymatroidal monomial ideal has linear quotients　119
5.4.5　Strongly shellable simplicial complexes　120
5.5　sCM simplicial complexes and sCM graded modules　121
5.6　Clique complex *， edge ideal I(G) and cover ideal Ic(G)　123
5. 7　Vertex-decomposable graphs a且d shellable graphs　124
5.8　Minimal verex covers and standard irredundant primary decomposition of I(G)　127
5.9　Cohen-Macaulay graphs and well-covered graphs　129
5.10　Shellable clutters　131
5.10.1　Clutters with the free vertex property　132
5.10.2　Chordal clutters　133
5.11　Some particular classes of graphs　133
5.11.1　Bipartite graphs　133
5.11.2　Boolean graphs are Cohen-Macaulay　138
5.11.3　Cactus graphs and classes of vertex-decomposable graphs　143
5.11.4　Cameron-Walker graphs　146
5.11.5　Chordal graphs　147
5.11.6　F-simplicial complexes and f-ideals of kind （n，d）　150
5.11.7　Gap-free graphs and related H-free graphs　174
5.11.8　Graphs whose complements are r-partite　176
5.11.9　Graph expansions and graph blow ups　185
5.11.10　Interlacing graphs * and triangular graphs *　188
5.11.11 Vertex clique-whiskered graphs * and their generalizations *　189
5.11.12　1-decomposable graphs　202
Chapter 6　Shellable Simplicial Complexes from Posets　204
6.1　Preliminaries　204
6.2　A bounded， locally upper-semimodular poset is pure shellable　205
6.3　EL-labeling of a poset and EL-shellable graded posets　209
6.4　Admissible lattices and SL-shellable poset　213
6.5　CL-shellable poset and recursive atom orderings　215
6.5.1　Rooted interval and CL-shellable poset　216
6.5.2　Recursive atom orderings　218
Chapter7　Betti Numbers and Castelnuovo-Mumford Regularity　222
7.1　Calculating Betti numbers via the functor Tor　222
7.2　Polarization keeps the Betti numbers and regularity unchanged　224
7.3　Hochster’s Formula and other two reformulations　226
7.4　Graded Betti numbers of graphs: some general reults　230
7.5　Spli也table

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復形和Cohen-Macaulay性質(英文版) 節選

Chapter 1 Preliminaries on Cohen-Macaulay Rings and Modules All rings in this book are assumed to be commutative with identity element and， all modules are unital. An f.g. module RM denotes that M is finitely generated as a left R-module. In this section， we collect some fundamental results of commutative algebra and， efforts are made to make the exposition self-contained. We take the monograph [39] (in English）as the fundamental reference in commutative algebra， and one can also use [119] (in Chinese）for some basic facts of commutative rings. This section also contains two particular topics. Recall that a ring R is called Cohen-Macaulay (often abbreviated as CM for brevity) if the local ring Rp is CohenMacaulay，i.e， depth( Pp) = dim (Rp) holds for every prime (or equivalently， maximal) ideal P of R. For any field ，set and . Then for any graded proper ideal I of S， there is a single condition on R =: S/ I， i.e.， depth(m/1，R) = dimR， which we call grnded Cohen-Macaulay. In Therorem 1.61， we will give a direct proof to the following expected facts: The graded ring S/ I is graded Cohen-Macaulay if and only if it is Cohen-Macaulay. In the final part， we include an elementary approach of Schenzel’s view on sequentially Cohen-Macaulay modules， in which the key tool is the saturated submodule of nM related to an ideal I of R，see [39， Section 3.6] for a general discussion on the importance of the local cohomology module . 1.1　Jacobson radical and NAK Lemma For a ring R， the Jacobson radical J(R) is defined to be the intersection of all maximal left (or equivalently， right) ideals of R. Note that J ( R) contains no nonzero idempotent elements of R， and where U(R) consists of all units of R. Lemma 1.1 (NAK Lemma)　Let M be an f.g. module over α ring R. For an ideal I of R，if and IM= M，then M = 0. Proof　Let and let . It follows from that α ＝ Aα for some . Then we have where and， denotes the adjoint matrix of a matrix B. Since 1 + b is invertible in R，It follows that ， thus M = 0. Corollary 1.2　Let M be an f. g. module over a ring R and let N be a submodule of M. For an ideal I of R， if and ， then. Thus the module J(R)M is a superfluous (or alternatively， a small) submodule of an f.g. module M. Note that Nakayama’s Lemma follows easily from the following Cayley-Hamilton Theorem: Theorem 1.3　Let I be an ideal of R， and assume that α module RM is generated by n elements. Then for each endomorphism of M with， there exist elements in such that the polynomial annihilates ． Proof　For any ，set xm ＝ .Assume RM= . Then there exists a matrix such that ， where .Since R is assumed to be commutative， we have where denotes the adjoint matrix of Bin Mn(R). Then the polynomial annihilates*， where * holds for all i. Cayley-Hamilton Theorem can be applied to verify the following: Corollary 1.4　Let R be a commutative ring and RM an f.g. module. Then ( 1) Any surjective endomorphism of M is an isomorphism. (2) A commutative ring has the Invariance Basis Nwmber， i. e.， any module isomorphism implies n = m. Note that a non.commutative Noetherian ring also has the properties in Corollary 1.4. The proof to this fact is left as an exercise. Remark　NAK is the abbreviation of Nakayama-Azumaya-Krull. It is also the first three letters of Nakayama. 1.2 Modules with finite lengths Recall that a module RM has finite length if and only if M is both Artinian and Noetherian. For a prime ideal P of R， recall that if and only if ann . Recall that Supp(P) consists of prime ideals P such that . The following proposition implies the well-known Jordan-Holder Theorem: Lemma 1.5 Let (l.2.1) be a composition series of the module M， i.e.， each quotient module is simple. Let Then (1) For any maximal ideal Q of R ， if and only if . Thus Supp(M) and hence， it is independent of the choice of the composition series. (2) For distinct maximal ideals P and Q of R， we have *. (3) For any *， the number of Mi such that * is the length of Mp as an Rp-module. Proof (1) Let * and assume*Then we have *and *. Take* such that *. Then we have*， hence*. Thus *， and it follows from * that MQ = 0. On the other hand， if *， assume* for some i. Since R/Q is a field， we have*. Thus*， hence*. This shows*， and completes the proof to (1). Note that * holds for any maximal ideal P distinct to Q. (2) and (3): If M is a simple module，then*， where P = ann(M). Then Mp = M and MQ = 0 for any maximal ideal *. Now assume length(M) = n > 1. Then for any maximal ideal P*A， we have an Rp-module sequence: Note that each M3/M3-1 is a simple module， and while the latter holds if and only if*. Thus the length of the Rpmodule Mp is the number of Mk such that*. In particular， it follows from the filtration (1.2.2) that (Mp )Q = 0 holds for distinct maximal ideals P and Q. Theorem 1.6 Let M be a module with

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